Topological data analysis (TDA) is field that started only two decades ago and has already shown promise both in theory and in applications. The goal of TDA is to quantify the shape of data in a manner that is concise and robust using concepts from algebraic topology. Persistent homology, arguably the most popular tool from TDA, studies the shape of a filtered space by watching how its homology changes. The output of persistent homology is a persistence diagram, which encodes information about the changing homology. Persistent homology has shown success in various application areas; one ever growing area of study in this field is time series analysis. Nonlinear time series analysis is a research field in and of itself that aims to capture structure in time series data, however, it lacks theoretically justified tools to analyze the resulting structure. Persistent homology comes with a solid theoretical framework, is robust to noise, and quantifies the same type of structure as appears in time series data. Thus combining tools from time series analysis and TDA provides a new approach to analyze and quantify behavior in time series data. One field where time series are prevalent is dynamical systems, since a time series arises from a projection of a solution to a system. Specifically, given a time series, Takens' theorem can be leveraged to embed the time series as a point cloud in a higher dimensional space, where this point cloud is a sampling of the full state space. Then for each time series, persistent homology can be computed on the embedding. The result is a persistence diagram for each time series. The question then becomes how do we analyze this collection of persistence diagrams to learn something about the original time series data? Many people have developed methods to answer this question, through methods such as machine learning or statistics. This dissertation provides several new methods leveraging tools from both TDA and nonlinear time series analysis to study time varying data.
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